A071825 a(n) = (number of x with largest prime factor of the form 4k+3 less than or equal to n) minus (number of x with largest prime factor of the form 4k+1 less than or equal to n).

0, 0, 1, 1, 0, 1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 6, 7, 6, 5, 6, 6, 7, 6, 7, 8, 7, 8, 7, 6, 5, 6, 7, 8, 7, 8, 9, 10, 11, 10, 9, 8, 7, 8, 9, 10, 11, 10, 11, 10, 9, 10, 11, 11, 10, 11, 12, 11, 12, 13, 14, 15, 14, 13, 12, 13, 14, 13, 14, 13, 14, 13, 14, 15, 14, 15, 14, 15

Offset: 1

Author

Benoit Cloitre, Jun 07 2002

Keywords

• easy
• nonn
• look

Comments

Weak conjecture: sequence gives positive values only; strong conjecture: for n large enough a(n) > 2*sqrt(n).

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Robert Israel, Log-log plot of (n, a(n)) for 1 <= n <= 10^6

Crossrefs

Cf. A071823, A071824.

Programs

Maple

f:= proc(n) local t; t:= max(numtheory:-factorset(n)) mod 4; if t = 3 then 1 elif t= 1 then -1 else 0 fi end proc:
ListTools:-PartialSums(map(f, [$1..100])); # Robert Israel, Nov 05 2024

PARI

for(n=1,200,print1(sum(i=2,n,if((component(component(factor(i),1),omega(i))-3)%4,0,1))-sum(i=2,n,if((component(component(factor(i),1),omega(i))-1)%4,0,1)),","))

Formula

a(n) = Card( x<=n : A006530(x) == 3 mod 4) - Card( x<=n : A006530(x) == 1 mod 4)

a(n) = A071823(n) - A071824(n). - Sean A. Irvine, Aug 16 2024

Extensions

Missing a(1)=0 inserted by Sean A. Irvine, Aug 16 2024